Courant Algebroidsand Generalizations of Geometry
Peter Bouwknegt (1,2)
(1) Department of MathematicsMathematical Sciences Institute
(2) Department of Theoretical PhysicsResearch School of Physics and Engineering
The Australian National UniversityCanberra, AUSTRALIA
String-Math 2011, UPenn9 June 2011
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Introduction
Generalizations of geometry
TM ⊕ T ∗M (type I or type II in absence of RR fluxes)TM ⊕ T ∗M ⊕ S± (type IIA/IIB with RR fluxes)TM ⊕ T ∗M ⊕G (type I + YM, heterotic)TM ⊕ ∧2T ∗M ⊕ . . . (M-theory)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Introduction
Generalizations of geometryTM ⊕ T ∗M (type I or type II in absence of RR fluxes)
TM ⊕ T ∗M ⊕ S± (type IIA/IIB with RR fluxes)TM ⊕ T ∗M ⊕G (type I + YM, heterotic)TM ⊕ ∧2T ∗M ⊕ . . . (M-theory)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Introduction
Generalizations of geometryTM ⊕ T ∗M (type I or type II in absence of RR fluxes)TM ⊕ T ∗M ⊕ S± (type IIA/IIB with RR fluxes)
TM ⊕ T ∗M ⊕G (type I + YM, heterotic)TM ⊕ ∧2T ∗M ⊕ . . . (M-theory)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Introduction
Generalizations of geometryTM ⊕ T ∗M (type I or type II in absence of RR fluxes)TM ⊕ T ∗M ⊕ S± (type IIA/IIB with RR fluxes)TM ⊕ T ∗M ⊕G (type I + YM, heterotic)
TM ⊕ ∧2T ∗M ⊕ . . . (M-theory)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Introduction
Generalizations of geometryTM ⊕ T ∗M (type I or type II in absence of RR fluxes)TM ⊕ T ∗M ⊕ S± (type IIA/IIB with RR fluxes)TM ⊕ T ∗M ⊕G (type I + YM, heterotic)TM ⊕ ∧2T ∗M ⊕ . . . (M-theory)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Outline of the talk
Review of generalized geometryBeyond generalized geometry
B2n-geometryM-geometry
T-duality and generalized geometry
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Outline of the talk
Review of generalized geometry
Beyond generalized geometryB2n-geometryM-geometry
T-duality and generalized geometry
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Outline of the talk
Review of generalized geometryBeyond generalized geometry
B2n-geometryM-geometry
T-duality and generalized geometry
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Outline of the talk
Review of generalized geometryBeyond generalized geometry
B2n-geometryM-geometry
T-duality and generalized geometry
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Outline of the talk
Review of generalized geometryBeyond generalized geometry
B2n-geometryM-geometry
T-duality and generalized geometry
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Generalized geometry [Hitchin, Gualtieri, Cavalcanti]
Replace structures on TM (such as [ , ], ıx , Lx , d , ...) by similarstructures on E = TM ⊕ T ∗M
Bilinear form on sections x + ξ ∈ Γ(TM ⊕ T ∗M)
〈x + ξ, y + η〉 = 12(ıxη + ıyξ)
(Dorfman) Bracket
(x + ξ) (y + η) = [x , y ] + Lxη − ıydξ
Clifford algebra
γx+ξ, γy+η = 2〈x + ξ, y + η〉
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Generalized geometry [Hitchin, Gualtieri, Cavalcanti]
Replace structures on TM (such as [ , ], ıx , Lx , d , ...) by similarstructures on E = TM ⊕ T ∗M
Bilinear form on sections x + ξ ∈ Γ(TM ⊕ T ∗M)
〈x + ξ, y + η〉 = 12(ıxη + ıyξ)
(Dorfman) Bracket
(x + ξ) (y + η) = [x , y ] + Lxη − ıydξ
Clifford algebra
γx+ξ, γy+η = 2〈x + ξ, y + η〉
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Generalized geometry [Hitchin, Gualtieri, Cavalcanti]
Replace structures on TM (such as [ , ], ıx , Lx , d , ...) by similarstructures on E = TM ⊕ T ∗M
Bilinear form on sections x + ξ ∈ Γ(TM ⊕ T ∗M)
〈x + ξ, y + η〉 = 12(ıxη + ıyξ)
(Dorfman) Bracket
(x + ξ) (y + η) = [x , y ] + Lxη − ıydξ
Clifford algebra
γx+ξ, γy+η = 2〈x + ξ, y + η〉
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Generalized geometry [Hitchin, Gualtieri, Cavalcanti]
Replace structures on TM (such as [ , ], ıx , Lx , d , ...) by similarstructures on E = TM ⊕ T ∗M
Bilinear form on sections x + ξ ∈ Γ(TM ⊕ T ∗M)
〈x + ξ, y + η〉 = 12(ıxη + ıyξ)
(Dorfman) Bracket
(x + ξ) (y + η) = [x , y ] + Lxη − ıydξ
Clifford algebra
γx+ξ, γy+η = 2〈x + ξ, y + η〉
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Generalized Geometry (cont’d)
Clifford module Ω•(M)
γx+ξ · ω = ıxω + ξ ∧ ω
De-Rham differential on Ω•(M)
d : Ωk (M)→ Ωk+1(M)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Generalized Geometry (cont’d)
Clifford module Ω•(M)
γx+ξ · ω = ıxω + ξ ∧ ω
De-Rham differential on Ω•(M)
d : Ωk (M)→ Ωk+1(M)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Symmetries
Symmetries of 〈 , 〉 given by sections of the adjoint bundle
∧2E ∼= ∧2TM ⊕ End(TM)⊕ ∧2T ∗M
[They form the group O(n,n)]
In particular, we have the so called B-transform, for b ∈ Ω2(M)
eb · (x + ξ) = x + (ξ + ıxb)
b · (x + ξ) = ıxb (infinitesimally)
We have
eb · ((x + ξ) (y + η)) = eb · (x + ξ) eb · (y + η) + ıx ıydb
Symmetries of the Dorfman bracket are Diff(M) n Ω2cl(M)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Symmetries
Symmetries of 〈 , 〉 given by sections of the adjoint bundle
∧2E ∼= ∧2TM ⊕ End(TM)⊕ ∧2T ∗M
[They form the group O(n,n)]In particular, we have the so called B-transform, for b ∈ Ω2(M)
eb · (x + ξ) = x + (ξ + ıxb)
b · (x + ξ) = ıxb (infinitesimally)
We have
eb · ((x + ξ) (y + η)) = eb · (x + ξ) eb · (y + η) + ıx ıydb
Symmetries of the Dorfman bracket are Diff(M) n Ω2cl(M)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Symmetries
Symmetries of 〈 , 〉 given by sections of the adjoint bundle
∧2E ∼= ∧2TM ⊕ End(TM)⊕ ∧2T ∗M
[They form the group O(n,n)]In particular, we have the so called B-transform, for b ∈ Ω2(M)
eb · (x + ξ) = x + (ξ + ıxb)
b · (x + ξ) = ıxb (infinitesimally)
We have
eb · ((x + ξ) (y + η)) = eb · (x + ξ) eb · (y + η) + ıx ıydb
Symmetries of the Dorfman bracket are Diff(M) n Ω2cl(M)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Symmetries
Symmetries of 〈 , 〉 given by sections of the adjoint bundle
∧2E ∼= ∧2TM ⊕ End(TM)⊕ ∧2T ∗M
[They form the group O(n,n)]In particular, we have the so called B-transform, for b ∈ Ω2(M)
eb · (x + ξ) = x + (ξ + ıxb)
b · (x + ξ) = ıxb (infinitesimally)
We have
eb · ((x + ξ) (y + η)) = eb · (x + ξ) eb · (y + η) + ıx ıydb
Symmetries of the Dorfman bracket are Diff(M) n Ω2cl(M)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Twisting
This suggest the introduction of a twisted Dorfman bracket, withH ∈ Ω3(M), dH = 0
(x + ξ) H (y + η) = [x , y ] + Lxη − ıydξ + ıx ıyH
such that
eb · ((x + ξ) H (y + η)) =(
eb · (x + ξ))H+db
(eb · (y + η)
)and a twisted differential
dHω = dω + H ∧ ω
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Twisting
This suggest the introduction of a twisted Dorfman bracket, withH ∈ Ω3(M), dH = 0
(x + ξ) H (y + η) = [x , y ] + Lxη − ıydξ + ıx ıyH
such that
eb · ((x + ξ) H (y + η)) =(
eb · (x + ξ))H+db
(eb · (y + η)
)
and a twisted differential
dHω = dω + H ∧ ω
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Twisting
This suggest the introduction of a twisted Dorfman bracket, withH ∈ Ω3(M), dH = 0
(x + ξ) H (y + η) = [x , y ] + Lxη − ıydξ + ıx ıyH
such that
eb · ((x + ξ) H (y + η)) =(
eb · (x + ξ))H+db
(eb · (y + η)
)and a twisted differential
dHω = dω + H ∧ ω
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Properties of the (twisted) Dorfman bracket
Properties (for A,B,C ∈ ΓE , f ∈ C∞(M))(i) A (B C) = (A B) C + B (A C)
(ii) A (fB) = f (A B) + (ρ(A)f )B
The Courant bracket is defined as the anti-symmetrization
[[A,B]] =12
(A B − B A)
or, conversely,A B = [[A,B]] + d〈A,B〉
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Properties of the (twisted) Dorfman bracket
Properties (for A,B,C ∈ ΓE , f ∈ C∞(M))(i) A (B C) = (A B) C + B (A C)
(ii) A (fB) = f (A B) + (ρ(A)f )B
The Courant bracket is defined as the anti-symmetrization
[[A,B]] =12
(A B − B A)
or, conversely,A B = [[A,B]] + d〈A,B〉
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Dorfman bracket as a derived bracket
Recall the usual Cartan relations
ıx , ıy = 0d , ıx = Lx
[Lx , ıy ] = ı[x ,y ]
[Lx ,Ly ] = L[x ,y ]
[d ,Lx ] = 0
Here we have the analogue
γA, γB = 2〈A,B〉dH , γA = LA
[LA, γB] = γAB
[LA,LB] = LAB = L[[A,B]]
[dH ,LA] = 0
where Lx+ξω = Lxω + (dξ + ıxH) ∧ ω
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Dorfman bracket as a derived bracket
Recall the usual Cartan relations
ıx , ıy = 0d , ıx = Lx
[Lx , ıy ] = ı[x ,y ]
[Lx ,Ly ] = L[x ,y ]
[d ,Lx ] = 0
Here we have the analogue
γA, γB = 2〈A,B〉dH , γA = LA
[LA, γB] = γAB
[LA,LB] = LAB = L[[A,B]]
[dH ,LA] = 0
where Lx+ξω = Lxω + (dξ + ıxH) ∧ ωPeter Bouwknegt Courant Algebroids and Generalizations of Geometry
Leibniz algebroids
DefinitionA Leibniz algebroid (E , , ρ) is a vector bundle E → M, with acomposition (Leibniz/Loday bracket) on ΓE , and a morphismof vector bundles ρ : E → TM (anchor) such that(L1) A (B C) = (A B) C + B (A C)
(L2) ρ(A B) = [ρ(A), ρ(B)]
(L3) A (fB) = f (A B) + (ρ(A)f )B
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Courant algebroids
DefinitionA Courant algebroid (E , , 〈 , 〉, ρ) is a vector bundle E → M,with a composition on ΓE , a morphism of vector bundlesρ : E → TM, and a field of nondegenerate bilinear forms 〈 , 〉on ΓE such that
(C1) A (B C) = (A B) C + B (A C)
(C2) ρ(A)〈B,C〉 = 〈A,B C + C B〉(C3) ρ(A)〈B,C〉 = 〈A B,C〉+ 〈B,A C〉
It follows that (E , , ρ) is a Leibniz algebroid. Moreover
A B + B A = 2D〈A,B〉
where D = 12ρ∗d : C∞(M)→ ΓE , i.e. 〈Df ,A〉 = 1
2ρ(A)f
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Courant algebroids
DefinitionA Courant algebroid (E , , 〈 , 〉, ρ) is a vector bundle E → M,with a composition on ΓE , a morphism of vector bundlesρ : E → TM, and a field of nondegenerate bilinear forms 〈 , 〉on ΓE such that
(C1) A (B C) = (A B) C + B (A C)
(C2) ρ(A)〈B,C〉 = 〈A,B C + C B〉(C3) ρ(A)〈B,C〉 = 〈A B,C〉+ 〈B,A C〉
It follows that (E , , ρ) is a Leibniz algebroid. Moreover
A B + B A = 2D〈A,B〉
where D = 12ρ∗d : C∞(M)→ ΓE , i.e. 〈Df ,A〉 = 1
2ρ(A)f
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Exact Courant algebroids
An exact Courant algebroid E is a Courant algebroid that fits inthe exact sequence
0 −−−−→ T ∗Mρ∗−−−−→ E
ρ−−−−→ TM −−−−→ 0
Such a Courant algebroid admits an isotropic splittings : TM → E , which allows us to identify E ∼= TM ⊕ T ∗M. Thecomposition on x + ξ ∈ Γ(TM ⊕T ∗M) is uniquely determined by
H(x , y , z) = 〈x y , z〉
It turns out H ∈ Ω3cl(M)
Theorem (Ševera)Equivalence classes of exact Courant algebroids are in 1–1correspondence with H3(M,R).
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Exact Courant algebroids
An exact Courant algebroid E is a Courant algebroid that fits inthe exact sequence
0 −−−−→ T ∗Mρ∗−−−−→ E
ρ−−−−→ TM −−−−→ 0
Such a Courant algebroid admits an isotropic splittings : TM → E , which allows us to identify E ∼= TM ⊕ T ∗M. Thecomposition on x + ξ ∈ Γ(TM ⊕T ∗M) is uniquely determined by
H(x , y , z) = 〈x y , z〉
It turns out H ∈ Ω3cl(M)
Theorem (Ševera)Equivalence classes of exact Courant algebroids are in 1–1correspondence with H3(M,R).
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Exact Courant algebroids
An exact Courant algebroid E is a Courant algebroid that fits inthe exact sequence
0 −−−−→ T ∗Mρ∗−−−−→ E
ρ−−−−→ TM −−−−→ 0
Such a Courant algebroid admits an isotropic splittings : TM → E , which allows us to identify E ∼= TM ⊕ T ∗M. Thecomposition on x + ξ ∈ Γ(TM ⊕T ∗M) is uniquely determined by
H(x , y , z) = 〈x y , z〉
It turns out H ∈ Ω3cl(M)
Theorem (Ševera)Equivalence classes of exact Courant algebroids are in 1–1correspondence with H3(M,R).
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
B2n-geometry [Baraglia]
We now consider the vector bundle
E = TM ⊕ 1⊕ T ∗M
with nondegenerate bilinear form
〈x + f + ξ, y + g + η〉 = 12(ıxη + ıyξ) + fg
Dorfman bracket
(x + f +ξ)(y +g +η) = [x , y ]+(x(g)−y(f ))+Lxη− ıydξ+2gdf
and anchor mapρ(x + f + ξ) = x
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
B2n-geometry [Baraglia]
We now consider the vector bundle
E = TM ⊕ 1⊕ T ∗M
with nondegenerate bilinear form
〈x + f + ξ, y + g + η〉 = 12(ıxη + ıyξ) + fg
Dorfman bracket
(x + f +ξ)(y +g +η) = [x , y ]+(x(g)−y(f ))+Lxη− ıydξ+2gdf
and anchor mapρ(x + f + ξ) = x
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
B2n-geometry [Baraglia]
We now consider the vector bundle
E = TM ⊕ 1⊕ T ∗M
with nondegenerate bilinear form
〈x + f + ξ, y + g + η〉 = 12(ıxη + ıyξ) + fg
Dorfman bracket
(x + f +ξ)(y +g +η) = [x , y ]+(x(g)−y(f ))+Lxη− ıydξ+2gdf
and anchor mapρ(x + f + ξ) = x
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
B2n-geometry (cont’d)
Adjoint bundle
∧2E = ∧2TM ⊕ TM ⊕ End(TM)⊕ T ∗M ⊕ ∧2T ∗M
In particular
a · (x + f + ξ) = ıxa− fa , a ∈ Ω1
b · (x + f + ξ) = ıxb , b ∈ Ω2
Note[a1,a2] = a1 ∧ a2
Symmetries of the Dorfman bracket iff da = 0 = db.
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
B2n-geometry (cont’d)
Adjoint bundle
∧2E = ∧2TM ⊕ TM ⊕ End(TM)⊕ T ∗M ⊕ ∧2T ∗M
In particular
a · (x + f + ξ) = ıxa− fa , a ∈ Ω1
b · (x + f + ξ) = ıxb , b ∈ Ω2
Note[a1,a2] = a1 ∧ a2
Symmetries of the Dorfman bracket iff da = 0 = db.
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
B2n-geometry (cont’d)
Adjoint bundle
∧2E = ∧2TM ⊕ TM ⊕ End(TM)⊕ T ∗M ⊕ ∧2T ∗M
In particular
a · (x + f + ξ) = ıxa− fa , a ∈ Ω1
b · (x + f + ξ) = ıxb , b ∈ Ω2
Note[a1,a2] = a1 ∧ a2
Symmetries of the Dorfman bracket iff da = 0 = db.
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
B2n-geometry (cont’d)
Adjoint bundle
∧2E = ∧2TM ⊕ TM ⊕ End(TM)⊕ T ∗M ⊕ ∧2T ∗M
In particular
a · (x + f + ξ) = ıxa− fa , a ∈ Ω1
b · (x + f + ξ) = ıxb , b ∈ Ω2
Note[a1,a2] = a1 ∧ a2
Symmetries of the Dorfman bracket iff da = 0 = db.
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
B2n-geometry (cont’d)
Consider F ∈ Ω2(M), H ∈ Ω3(M), and a twisted Dorfmanbracket
(x + f + ξ) (y + g + η) = [x , y ] + (x(g)− y(f )) + ıx ıyF+ Lxη − ıydξ + 2gdf + ıx ıyH + ıxFg − ıyFf
This defines a Courant algebroid provided
dF = 0dH + F ∧ F = 0
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
B2n-geometry (cont’d)
Consider F ∈ Ω2(M), H ∈ Ω3(M), and a twisted Dorfmanbracket
(x + f + ξ) (y + g + η) = [x , y ] + (x(g)− y(f )) + ıx ıyF+ Lxη − ıydξ + 2gdf + ıx ıyH + ıxFg − ıyFf
This defines a Courant algebroid provided
dF = 0dH + F ∧ F = 0
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
B2n-geometry (cont’d)
The bracket can be obtained as a derived bracket using
dF ,H ω = dω + (−1)|ω|F ∧ ω + H ∧ ω
andγx+f +ξ · ω = ıxω + (−1)|ω|fω + ξ ∧ ω
Note
dF ,H2 = 0
γx+f +ξ, γy+g+η
= 2〈x + f + ξ, y + g + η〉
iff F and H satisfy the Bianchi identities as before.This is an example of non-exact transitive Courant algebroid
T ∗M −−−−→ TM ⊕ 1⊕ T ∗M −−−−→ TM −−−−→ 0
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
B2n-geometry (cont’d)
The bracket can be obtained as a derived bracket using
dF ,H ω = dω + (−1)|ω|F ∧ ω + H ∧ ω
andγx+f +ξ · ω = ıxω + (−1)|ω|fω + ξ ∧ ω
Note
dF ,H2 = 0
γx+f +ξ, γy+g+η
= 2〈x + f + ξ, y + g + η〉
iff F and H satisfy the Bianchi identities as before.
This is an example of non-exact transitive Courant algebroid
T ∗M −−−−→ TM ⊕ 1⊕ T ∗M −−−−→ TM −−−−→ 0
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
B2n-geometry (cont’d)
The bracket can be obtained as a derived bracket using
dF ,H ω = dω + (−1)|ω|F ∧ ω + H ∧ ω
andγx+f +ξ · ω = ıxω + (−1)|ω|fω + ξ ∧ ω
Note
dF ,H2 = 0
γx+f +ξ, γy+g+η
= 2〈x + f + ξ, y + g + η〉
iff F and H satisfy the Bianchi identities as before.This is an example of non-exact transitive Courant algebroid
T ∗M −−−−→ TM ⊕ 1⊕ T ∗M −−−−→ TM −−−−→ 0
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Transitive Courant Algebroids [Chen-Stiénon-Xu]
Transitive Courant algebroids are of the form
E = TM ⊕G⊕ T ∗M
where G = kerρ/(kerρ)⊥ is a bundle of Lie algebras withbracket
[r , s]G = prG(r s)
andρ(x + r + ξ) = x
Suppose
〈x + r + ξ, y + s + η〉 = 12 (ıxη + ıyξ) + 〈r , s〉G
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Transitive Courant Algebroids [Chen-Stiénon-Xu]
Transitive Courant algebroids are of the form
E = TM ⊕G⊕ T ∗M
where G = kerρ/(kerρ)⊥ is a bundle of Lie algebras withbracket
[r , s]G = prG(r s)
andρ(x + r + ξ) = x
Suppose
〈x + r + ξ, y + s + η〉 = 12 (ıxη + ıyξ) + 〈r , s〉G
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Transitive Courant Algebroids (cont’d)
Then the Dorfman bracket on E is completely determined by
H(x , y , z) = 〈prT∗M(x y), z〉R(x , y) = prG(x y)
∇x r = prG(x r) .
It turns out H ∈ Ω3(M), R ∈ Ω2(M, g), and ∇x a TM-connectionon G
Namely
(x + r + ξ) (y + s + η) = [x , y ]
− ıx ıyR + [r , s]G +∇xs −∇y r− ıx ıyH + 〈s,d∇r〉G + Lxη − ıydξ + 〈ıxR, s〉G − 〈ıyR, r〉G
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Transitive Courant Algebroids (cont’d)
Then the Dorfman bracket on E is completely determined by
H(x , y , z) = 〈prT∗M(x y), z〉R(x , y) = prG(x y)
∇x r = prG(x r) .
It turns out H ∈ Ω3(M), R ∈ Ω2(M, g), and ∇x a TM-connectionon GNamely
(x + r + ξ) (y + s + η) = [x , y ]
− ıx ıyR + [r , s]G +∇xs −∇y r− ıx ıyH + 〈s,d∇r〉G + Lxη − ıydξ + 〈ıxR, s〉G − 〈ıyR, r〉G
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Transitive Courant Algebroids (cont’d)
where we have defined a ‘twisted differential’d∇ : Ωk (M, ΓG)→ Ωk+1(M, ΓG) by
(d∇ω)(x0, . . . xk ) =k∑
i=0
(−1)i∇xiω(x0, . . . , xi , . . . , xk )
+∑
0≤i<j≤k
(−1)i+jω([xi , xj ], x0, . . . , . . . , xi , . . . , xj , . . . , xk )
[Note that d2∇ = 0 iff the curvature corresponding to ∇x
vanishes]
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Transitive Courant Algebroids (cont’d)
where
Lx〈r , s〉G = 〈∇x r , s〉G + 〈r ,∇xs〉G∇x [r , s]G = [∇x r , s]G + [r ,∇xs]G
d∇R = 0
d2∇r =
(∇x∇y −∇y∇x −∇[x ,y ]
)r = [R, r ]G ,
dH = 〈R ∧ R〉G .
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
M-geometry [Hull, Pacheco-Waldram, Baraglia]
In M-theory we have a 3-form C3 with
F4 = dC3
satisfyingdF4 = 0 (Bianchi)
andd(∗F4) + 1
2F4 ∧ F4 = 0 (e.o.m.)
After putting F7 = ∗F4 we have
d(F7 + 12C3 ∧ F4) = 0 , F7 + 1
2C3 ∧ F4 = dC6
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
M-geometry [Hull, Pacheco-Waldram, Baraglia]
In M-theory we have a 3-form C3 with
F4 = dC3
satisfyingdF4 = 0 (Bianchi)
andd(∗F4) + 1
2F4 ∧ F4 = 0 (e.o.m.)
After putting F7 = ∗F4 we have
d(F7 + 12C3 ∧ F4) = 0 , F7 + 1
2C3 ∧ F4 = dC6
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
M-geometry (cont’d)
Summarizing
F4 = dC3
F7 = dC6 − 12C3 ∧ F4
with
dF4 = 0
dF7 + 12F4 ∧ F4 = 0
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
M-geometry (cont’d)
Symmetries by z3 ∈ Ω3cl, z6 ∈ Ω6
cl
C′3 = C3 + z3
C′6 = C6 + z6 + 12C3 ∧ z3
Group law
(z3, z6) · (z ′3, z ′6) = (z3 + z ′3, z6 + z ′6 − 12z3 ∧ z ′3)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
M-geometry (cont’d)
The relevant bundle in this case is
E = TM ⊕ ∧2T ∗M ⊕ ∧5T ∗M
with Dorfman bracket
(x + a2 + a5) (y + b2 + b5) =
[x , y ] + Lxb2 − ıyda2 + Lxb5 − ıyda5 + da2 ∧ b2
The bracket is invariant under infinitesimal symmetriesgenerated by z3 ∈ Ω3
cl, z6 ∈ Ω6cl
z3 · (x + a2 + a5) = ıxz3 − z3 ∧ a2
z6 · (x + a2 + a5) = −ıxz6
.
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
M-geometry (cont’d)
The relevant bundle in this case is
E = TM ⊕ ∧2T ∗M ⊕ ∧5T ∗M
with Dorfman bracket
(x + a2 + a5) (y + b2 + b5) =
[x , y ] + Lxb2 − ıyda2 + Lxb5 − ıyda5 + da2 ∧ b2
The bracket is invariant under infinitesimal symmetriesgenerated by z3 ∈ Ω3
cl, z6 ∈ Ω6cl
z3 · (x + a2 + a5) = ıxz3 − z3 ∧ a2
z6 · (x + a2 + a5) = −ıxz6
.
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
M-geometry (cont’d)
It can be twisted by F4 ∈ Ω4, F7 ∈ Ω7
(x + a2 + a5) (y + b2 + b5) = [x , y ]
+ Lxb2 − ıyda2 + ıx ıyF4
+ Lxb5 − ıyda5 + da2 ∧ b2 + ıxF4 ∧ b2 + ıx ıyF7
and is a Leibniz algebroid iff
dF4 = 0
dF7 + 12F4 ∧ F4 = 0
Note that we have 〈 , 〉 : E ⊗ E → T ∗M ⊕ ∧4T ∗M
〈x + a2 + a5, y + b2 + b5〉 = (ıxb2 + ıya2) + (ıxb5 + ıya5 + a2∧b2)
(cf. notion of E-Courant algebroid [Chen-Liu-Sheng])
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
M-geometry (cont’d)
It can be twisted by F4 ∈ Ω4, F7 ∈ Ω7
(x + a2 + a5) (y + b2 + b5) = [x , y ]
+ Lxb2 − ıyda2 + ıx ıyF4
+ Lxb5 − ıyda5 + da2 ∧ b2 + ıxF4 ∧ b2 + ıx ıyF7
and is a Leibniz algebroid iff
dF4 = 0
dF7 + 12F4 ∧ F4 = 0
Note that we have 〈 , 〉 : E ⊗ E → T ∗M ⊕ ∧4T ∗M
〈x + a2 + a5, y + b2 + b5〉 = (ıxb2 + ıya2) + (ıxb5 + ıya5 + a2∧b2)
(cf. notion of E-Courant algebroid [Chen-Liu-Sheng])
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
M-geometry (cont’d)
It can be twisted by F4 ∈ Ω4, F7 ∈ Ω7
(x + a2 + a5) (y + b2 + b5) = [x , y ]
+ Lxb2 − ıyda2 + ıx ıyF4
+ Lxb5 − ıyda5 + da2 ∧ b2 + ıxF4 ∧ b2 + ıx ıyF7
and is a Leibniz algebroid iff
dF4 = 0
dF7 + 12F4 ∧ F4 = 0
Note that we have 〈 , 〉 : E ⊗ E → T ∗M ⊕ ∧4T ∗M
〈x + a2 + a5, y + b2 + b5〉 = (ıxb2 + ıya2) + (ıxb5 + ıya5 + a2∧b2)
(cf. notion of E-Courant algebroid [Chen-Liu-Sheng])
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality for principal S1-bundles
Suppose we have a pair (E ,H), consisting of a principal circlebundle
S1 −−−−→ E
π
yM
and a so-called H-flux H, a Cech 3-cocycle.
Topologically, E is classified by an element in F ∈ H2(M,Z)while H gives a class in H3(E ,Z)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality for principal S1-bundles (cont’d)
The T-dual of (E ,H) is given by the pair (E , H), where theprincipal S1-bundle
S1 −−−−→ E
π
yM
and the dual H-flux H ∈ H3(E ,Z), satisfy
F = π∗H , F = π∗H
where π∗ : H3(E ,Z)→ H2(M,Z), andπ∗ : H3(E ,Z)→ H2(M,Z) are the pushforward maps(‘integration over the S1-fiber’)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality for principal S1-bundles (cont’d)
E
π
E ×M E
p=π×1
p=1×π
M
E
π
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality for principal S1-bundles (cont’d)
The ambiguity in the choice of H is removed by requiring that
p∗H − p∗H ≡ 0
in H3(E ×M E ,Z), where E ×M E is the correspondence space
E ×M E = (x , x) ∈ E × E | π(x) = π(x)
Theorem (B-Evslin-Mathai)This T-duality gives rise to an isomorphism between the twistedcohomologies and twisted K-theories of (E ,H) and (E , H) (witha shift in degree by 1)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality for principal S1-bundles (cont’d)
The ambiguity in the choice of H is removed by requiring that
p∗H − p∗H ≡ 0
in H3(E ×M E ,Z), where E ×M E is the correspondence space
E ×M E = (x , x) ∈ E × E | π(x) = π(x)
Theorem (B-Evslin-Mathai)This T-duality gives rise to an isomorphism between the twistedcohomologies and twisted K-theories of (E ,H) and (E , H) (witha shift in degree by 1)
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality and generalized geometry
Given a principal circle bundle E with H-flux H ∈ Ω3cl(E)S1
S1 −−−−→ E
π
yM
H = H(3) + A ∧ H(2), F = dA
there exists a T-dual principal circle bundle
S1 −−−−→ E
π
yM
H = H(3) + A ∧ F , F = H(2) = dA
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality and generalized geometry
Given a principal circle bundle E with H-flux H ∈ Ω3cl(E)S1
S1 −−−−→ E
π
yM
H = H(3) + A ∧ H(2), F = dA
there exists a T-dual principal circle bundle
S1 −−−−→ E
π
yM
H = H(3) + A ∧ F , F = H(2) = dA
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality and generalized geometry (cont’d)
Theorem [B-Evslin-Mathai, Cavalcanti-Gualtieri](a) We have an isomorphism of differential complexes
τ : (Ω•(E)S1,dH)→ (Ω•(E)S1
,dH)
τ(Ω(k) + A ∧ Ω(k−1)) = −Ω(k−1) + A ∧ Ω(k)
τ dH = −dH τ
Hence, τ induces an isomorphism on twisted cohomology(b) We can identify (X + Ξ ∈ Γ(TE ⊕ T ∗E)S1
with a quadruple(x , f ; ξ,g)
X = x + f∂A , Ξ = ξ + gA
and define a map φ : Γ(TE ⊕ T ∗E)S1 → Γ(T E ⊕ T ∗E)S1
φ(x + f∂A + ξ + gA) = x + g∂A + ξ + f A
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality and generalized geometry (cont’d)
Theorem [B-Evslin-Mathai, Cavalcanti-Gualtieri](a) We have an isomorphism of differential complexes
τ : (Ω•(E)S1,dH)→ (Ω•(E)S1
,dH)
τ(Ω(k) + A ∧ Ω(k−1)) = −Ω(k−1) + A ∧ Ω(k)
τ dH = −dH τ
Hence, τ induces an isomorphism on twisted cohomology(b) We can identify (X + Ξ ∈ Γ(TE ⊕ T ∗E)S1
with a quadruple(x , f ; ξ,g)
X = x + f∂A , Ξ = ξ + gA
and define a map φ : Γ(TE ⊕ T ∗E)S1 → Γ(T E ⊕ T ∗E)S1
φ(x + f∂A + ξ + gA) = x + g∂A + ξ + f A
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality and generalized geometry (cont’d)
Theorem [B-Evslin-Mathai, Cavalcanti-Gualtieri](a) We have an isomorphism of differential complexes
τ : (Ω•(E)S1,dH)→ (Ω•(E)S1
,dH)
τ(Ω(k) + A ∧ Ω(k−1)) = −Ω(k−1) + A ∧ Ω(k)
τ dH = −dH τ
Hence, τ induces an isomorphism on twisted cohomology
(b) We can identify (X + Ξ ∈ Γ(TE ⊕ T ∗E)S1with a quadruple
(x , f ; ξ,g)X = x + f∂A , Ξ = ξ + gA
and define a map φ : Γ(TE ⊕ T ∗E)S1 → Γ(T E ⊕ T ∗E)S1
φ(x + f∂A + ξ + gA) = x + g∂A + ξ + f A
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality and generalized geometry (cont’d)
Theorem [B-Evslin-Mathai, Cavalcanti-Gualtieri](a) We have an isomorphism of differential complexes
τ : (Ω•(E)S1,dH)→ (Ω•(E)S1
,dH)
τ(Ω(k) + A ∧ Ω(k−1)) = −Ω(k−1) + A ∧ Ω(k)
τ dH = −dH τ
Hence, τ induces an isomorphism on twisted cohomology(b) We can identify (X + Ξ ∈ Γ(TE ⊕ T ∗E)S1
with a quadruple(x , f ; ξ,g)
X = x + f∂A , Ξ = ξ + gA
and define a map φ : Γ(TE ⊕ T ∗E)S1 → Γ(T E ⊕ T ∗E)S1
φ(x + f∂A + ξ + gA) = x + g∂A + ξ + f A
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality and generalized geometry (cont’d)
(b) The map φ is orthogonal wrt pairing on TE ⊕ T ∗E , henceτ induces an isomorphism of Clifford algebras
(c) For X + Ξ ∈ Γ((TE ⊕ T ∗E)S1) we have
τ(γX+Ξ · Ω) = γφ(X+Ξ) · τ(Ω)
Hence τ induces an isomorphism of Clifford modules(d) For Xi + Ξi ∈ Γ((TE ⊕ T ∗E)S1
) we have
φ ([[X1 + Ξ1,X2 + Ξ2]]H) = [[φ(X1 + Ξ1), φ(X2 + Ξ2)]]H
Hence φ gives a homomorphism of twisted Courantbrackets
It follows that T-duality acts naturally on generalized complexstructures, generalized Kähler structures, generalizedCalabi-Yau structures, ...
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality and generalized geometry (cont’d)
(b) The map φ is orthogonal wrt pairing on TE ⊕ T ∗E , henceτ induces an isomorphism of Clifford algebras
(c) For X + Ξ ∈ Γ((TE ⊕ T ∗E)S1) we have
τ(γX+Ξ · Ω) = γφ(X+Ξ) · τ(Ω)
Hence τ induces an isomorphism of Clifford modules(d) For Xi + Ξi ∈ Γ((TE ⊕ T ∗E)S1
) we have
φ ([[X1 + Ξ1,X2 + Ξ2]]H) = [[φ(X1 + Ξ1), φ(X2 + Ξ2)]]H
Hence φ gives a homomorphism of twisted Courantbrackets
It follows that T-duality acts naturally on generalized complexstructures, generalized Kähler structures, generalizedCalabi-Yau structures, ...
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality and generalized geometry (cont’d)
(b) The map φ is orthogonal wrt pairing on TE ⊕ T ∗E , henceτ induces an isomorphism of Clifford algebras
(c) For X + Ξ ∈ Γ((TE ⊕ T ∗E)S1) we have
τ(γX+Ξ · Ω) = γφ(X+Ξ) · τ(Ω)
Hence τ induces an isomorphism of Clifford modules
(d) For Xi + Ξi ∈ Γ((TE ⊕ T ∗E)S1) we have
φ ([[X1 + Ξ1,X2 + Ξ2]]H) = [[φ(X1 + Ξ1), φ(X2 + Ξ2)]]H
Hence φ gives a homomorphism of twisted Courantbrackets
It follows that T-duality acts naturally on generalized complexstructures, generalized Kähler structures, generalizedCalabi-Yau structures, ...
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality and generalized geometry (cont’d)
(b) The map φ is orthogonal wrt pairing on TE ⊕ T ∗E , henceτ induces an isomorphism of Clifford algebras
(c) For X + Ξ ∈ Γ((TE ⊕ T ∗E)S1) we have
τ(γX+Ξ · Ω) = γφ(X+Ξ) · τ(Ω)
Hence τ induces an isomorphism of Clifford modules(d) For Xi + Ξi ∈ Γ((TE ⊕ T ∗E)S1
) we have
φ ([[X1 + Ξ1,X2 + Ξ2]]H) = [[φ(X1 + Ξ1), φ(X2 + Ξ2)]]H
Hence φ gives a homomorphism of twisted Courantbrackets
It follows that T-duality acts naturally on generalized complexstructures, generalized Kähler structures, generalizedCalabi-Yau structures, ...
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality and generalized geometry (cont’d)
(b) The map φ is orthogonal wrt pairing on TE ⊕ T ∗E , henceτ induces an isomorphism of Clifford algebras
(c) For X + Ξ ∈ Γ((TE ⊕ T ∗E)S1) we have
τ(γX+Ξ · Ω) = γφ(X+Ξ) · τ(Ω)
Hence τ induces an isomorphism of Clifford modules(d) For Xi + Ξi ∈ Γ((TE ⊕ T ∗E)S1
) we have
φ ([[X1 + Ξ1,X2 + Ξ2]]H) = [[φ(X1 + Ξ1), φ(X2 + Ξ2)]]H
Hence φ gives a homomorphism of twisted Courantbrackets
It follows that T-duality acts naturally on generalized complexstructures, generalized Kähler structures, generalizedCalabi-Yau structures, ...
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
T-duality and generalized geometry (cont’d)
(b) The map φ is orthogonal wrt pairing on TE ⊕ T ∗E , henceτ induces an isomorphism of Clifford algebras
(c) For X + Ξ ∈ Γ((TE ⊕ T ∗E)S1) we have
τ(γX+Ξ · Ω) = γφ(X+Ξ) · τ(Ω)
Hence τ induces an isomorphism of Clifford modules(d) For Xi + Ξi ∈ Γ((TE ⊕ T ∗E)S1
) we have
φ ([[X1 + Ξ1,X2 + Ξ2]]H) = [[φ(X1 + Ξ1), φ(X2 + Ξ2)]]H
Hence φ gives a homomorphism of twisted Courantbrackets
It follows that T-duality acts naturally on generalized complexstructures, generalized Kähler structures, generalizedCalabi-Yau structures, ...
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Dimensionally reduced Dorfman bracket
The dimensionally reduced Dorfman bracket
(x1, f1;ξ1,g1) (x2, f2; ξ2,g2) =
([x1, x2], x1(f2)− x2(f1) + ıx1ıx2F ;
(Lx1ξ2 − ıx2dξ1) + ıx1ıx2H(3) + (df1g2 + f2dg1)
+ (g2ıx1F − g1ıx2F ) + (f2ıx1H(2) − f1ıx2H(2)),
x1(g2)− x2(g1) + ıx1ıx2H(2))
is that of the transitive Courant algebroidE = TM ⊕ (t⊕ t∗)⊕ T ∗M with R = −(F ,H(2)), H = −H(3) and〈 , 〉G the canonical pairing between t and t∗.
[Doubling of the Atiyah algebroid corresponding to the principalS1-bundle]
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Dimensionally reduced Dorfman bracket
The dimensionally reduced Dorfman bracket
(x1, f1;ξ1,g1) (x2, f2; ξ2,g2) =
([x1, x2], x1(f2)− x2(f1) + ıx1ıx2F ;
(Lx1ξ2 − ıx2dξ1) + ıx1ıx2H(3) + (df1g2 + f2dg1)
+ (g2ıx1F − g1ıx2F ) + (f2ıx1H(2) − f1ıx2H(2)),
x1(g2)− x2(g1) + ıx1ıx2H(2))
is that of the transitive Courant algebroidE = TM ⊕ (t⊕ t∗)⊕ T ∗M with R = −(F ,H(2)), H = −H(3) and〈 , 〉G the canonical pairing between t and t∗.[Doubling of the Atiyah algebroid corresponding to the principalS1-bundle]
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Generalization to principal torus bundles
We have
H = H(3) + Ai ∧ H i(2) + 1
2Ai ∧ Aj ∧ H ij(1) + 1
6Ai ∧ Aj ∧ Ak ∧ H ijk(0)
such that
dH =d + H(3) + F(2)i∂Ai
+ 12F(1)ij∂Ai ∧ ∂Aj + 1
6F(0)ijk∂Ai ∧ ∂Aj ∧ ∂Ak
+ Ai ∧ H i(2) + 1
2Ai ∧ Aj ∧ H ij(1) + 1
6Ai ∧ Aj ∧ Ak ∧ H ijk(0)
The F(1)ij and F(0)ijk are known as nongeometric fluxes
Theorem (B-Garretson-Kao)T-duality provides an isomorphism of (certain) Courantalgebroids
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Generalization to principal torus bundles
We have
H = H(3) + Ai ∧ H i(2) + 1
2Ai ∧ Aj ∧ H ij(1) + 1
6Ai ∧ Aj ∧ Ak ∧ H ijk(0)
such that
dH =d + H(3) + F(2)i∂Ai
+ 12F(1)ij∂Ai ∧ ∂Aj + 1
6F(0)ijk∂Ai ∧ ∂Aj ∧ ∂Ak
+ Ai ∧ H i(2) + 1
2Ai ∧ Aj ∧ H ij(1) + 1
6Ai ∧ Aj ∧ Ak ∧ H ijk(0)
The F(1)ij and F(0)ijk are known as nongeometric fluxes
Theorem (B-Garretson-Kao)T-duality provides an isomorphism of (certain) Courantalgebroids
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Generalization to principal torus bundles
We have
H = H(3) + Ai ∧ H i(2) + 1
2Ai ∧ Aj ∧ H ij(1) + 1
6Ai ∧ Aj ∧ Ak ∧ H ijk(0)
such that
dH =d + H(3) + F(2)i∂Ai + 12F(1)ij∂Ai ∧ ∂Aj + 1
6F(0)ijk∂Ai ∧ ∂Aj ∧ ∂Ak
+ Ai ∧ H i(2) + 1
2Ai ∧ Aj ∧ H ij(1) + 1
6Ai ∧ Aj ∧ Ak ∧ H ijk(0)
The F(1)ij and F(0)ijk are known as nongeometric fluxes
Theorem (B-Garretson-Kao)T-duality provides an isomorphism of (certain) Courantalgebroids
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Generalization to principal torus bundles
We have
H = H(3) + Ai ∧ H i(2) + 1
2Ai ∧ Aj ∧ H ij(1) + 1
6Ai ∧ Aj ∧ Ak ∧ H ijk(0)
such that
dH =d + H(3) + F(2)i∂Ai + 12F(1)ij∂Ai ∧ ∂Aj + 1
6F(0)ijk∂Ai ∧ ∂Aj ∧ ∂Ak
+ Ai ∧ H i(2) + 1
2Ai ∧ Aj ∧ H ij(1) + 1
6Ai ∧ Aj ∧ Ak ∧ H ijk(0)
The F(1)ij and F(0)ijk are known as nongeometric fluxes
Theorem (B-Garretson-Kao)T-duality provides an isomorphism of (certain) Courantalgebroids
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
Generalization to principal torus bundles
We have
H = H(3) + Ai ∧ H i(2) + 1
2Ai ∧ Aj ∧ H ij(1) + 1
6Ai ∧ Aj ∧ Ak ∧ H ijk(0)
such that
dH =d + H(3) + F(2)i∂Ai + 12F(1)ij∂Ai ∧ ∂Aj + 1
6F(0)ijk∂Ai ∧ ∂Aj ∧ ∂Ak
+ Ai ∧ H i(2) + 1
2Ai ∧ Aj ∧ H ij(1) + 1
6Ai ∧ Aj ∧ Ak ∧ H ijk(0)
The F(1)ij and F(0)ijk are known as nongeometric fluxes
Theorem (B-Garretson-Kao)T-duality provides an isomorphism of (certain) Courantalgebroids
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry
THANKS
Peter Bouwknegt Courant Algebroids and Generalizations of Geometry